A posteriori error estimates for finite volume and mixed finite element discretizations of convection–diffusion–reaction equations

Abstract

We present in this paper a unified framework for a posteriori error estimation in the finite volume and lowest-order Raviart–Thomas mixed finite element methods. We consider convection–diffusion–reaction equations on simplicial meshes in two or three space dimensions and pay a particular attention to inhomogeneous and anisotropic diffusion–dispersion tensors and to convection dominance, in which case upwind-weighted schemes are considered. Our estimates are derived in the energy (semi-)norm for a locally postprocessed approximate solution preserving the finite volume/mixed finite element fluxes and are of residual type. They give a global upper bound on the error and are fully computable in the sense that all occurring constants are evaluated explicitly, so that they can serve both as indicators for adaptive refinement or for the actual control of the error. Local efficiency and semi-robustness in the sense that the local efficiency constant only depends on local variations in the inhomogeneities and anisotropies and becomes optimal as the local Peclet number gets sufficiently small can also be shown. Moreover, passing from their local to global computation, our estimates become asymptotically exact and asymptotically fully robust with respect to inhomogeneities and anisotropies. We finally present numerical experiments confirming their accuracy and briefly compare the results for the two methods.